This is an introduction to set theory and logic that starts completely from scratch. The text is accompanied by many methodological remarks and explanations.

The hope of this work is that these notes will help to convince more mathematicians that non-standard analysis can make a useful addition to the tools of their trade.

In this account, we explain the meaning of a priori voting power and outline how it is measured. We distinguish two intuitive notions as to what voting power means, leading to two approaches to measuring it. We discuss some philosophical and pragmatic objections, according to which a priori (as distinct from actual) voting power is worthless or inapplicable.

If K is an index of relative voting power for simple voting games, the bicameral postulate requires that the distribution of K -power within a voting assembly, as measured by the ratios of the powers of the voters, be independent of whether the assembly is viewed as a separate legislature or as one chamber of a bicameral system, provided that there are no voters common to both chambers. We argue that a reasonable index – if it is to be used (...) as a tool for analysing abstract, ‘uninhabited’ decision rules – should satisfy this postulate. We show that, among known indices, only the Banzhaf measure does so. Moreover, the Shapley–Shubik, Deegan–Packel and Johnston indices sometimes witness a reversal under these circumstances, with voter x ‘less powerful’ than y when measured in the simple voting game G1 , but ‘more powerful’ than y when G1 is ‘bicamerally joined’ with a second chamber G2 . Thus these three indices violate a weaker, and correspondingly more compelling, form of the bicameral postulate. It is also shown that these indices are not always co-monotonic with the Banzhaf index and that as a result they infringe another intuitively plausible condition – the price monotonicity condition. We discuss implications of these findings, in light of recent work showing that only the Shapley–Shubik index, among known measures, satisfies another compelling principle known as the bloc postulate. We also propose a distinction between two separate aspects of voting power: power as share in a fixed purse (P-power) and power as influence (I-power). (shrink)

LS Penrose was the first to propose a measure of voting power (which later came to be known as ‘the [absolute] Banzhaf index’). His limit theorem – which is implicit in Penrose (1952) and for which he gave no rigorous proof – says that, in simple weighted voting games, if the number of voters increases indefinitely while the quota is pegged at half the total weight, then – under certain conditions – the ratio between the voting powers (as measured by (...) him) of any two voters converges to the ratio between their weights. We conjecture that the theorem holds, under rather general conditions, for large classes of variously defined weighted voting games, other values of the quota, and other measures of voting power. We provide proofs for some special cases. (shrink)

We consider a singular event of the following form: in a simple voting game, a particular division of the voters resulted in a positive outcome. We propose a plausible measure that quantifies the causal contribution of any given voter to the outcome. This measure is based on a conceptual analysis due to Braham [1], but differs from his solution to the problem of measuring causality of singular events.

In the voting-power literature the rules of decision of the US Congress and the UN Security Council are widely misreported as though abstention amounts to a `no' vote. The hypothesis (proposed elsewhere) that this is due to a specific cause, theory-laden observation, is tested here by examining accounts of these rules in introductory textbooks on American Government and International Relations, where that putative cause does not apply. Our examination does not lead to a conclusive outcome regarding the hypothesis, but reveals (...) that the rules in question are also widely misreported in these textbooks. A second hypothesis---that the widespread misreporting is explicable by the relative rarity and unimportance of abstention in the two bodies concerned---is also tested and found to be untenable. (shrink)

We address simple voting games as mathematical objects in their own right, and study structures made up of these objects, rather than focusing on SVGs primarily as co-operative games. To this end it is convenient to employ the conceptual framework and language of category theory. This enables us to uncover the underlying unity of the basic operations involving SVGs.

L S Penrose’s Limit Theorem – which is implicit in Penrose [7, p. 72] and for which he gave no rigorous proof – says that, in simple weighted voting games, if the number of voters increases indefinitely and the relative quota is pegged, then – under certain conditions – the ratio between the voting powers of any two voters converges to the ratio between their weights. Lindner and Machover [4] prove some special cases of Penrose’s Limit Theorem. They give a (...) simple counter-example showing that the theorem does not hold in general even under the conditions assumed by Penrose; but they conjecture, in effect, that under rather general conditions it holds ‘almost always’ – that is with probability 1 – for large classes of weighted voting games, for various values of the quota, and with respect to several measures of voting power. We use simulation to test this conjecture. It is corroborated with respect to the Penrose–Banzhaf index for a quota of 50% but not for other values; with respect to the Shapley–Shubik index the conjecture is corroborated for all values of the quota. (shrink)

We analyse and evaluate the qualified majority (QM) decision rules for the Council of Ministers of the EU that are included in the Draft Constitution for Europe proposed by the European Convention [5]. We use a method similar to the one we used in [9] for the QM prescriptions made in the Treaty of Nice.

We analyse and evaluate the qualified majority (QM) decision rule for the Council of Ministers of the EU adopted at the EU Inter-Governmental Conference, Brussels, 18 June 2004 [1]. We compare this rule with the QM rule prescribed in the Treaty of Nice, and the rule included in the original draft Constitution proposed by the European Convention in July 2003. We use a method similar to the one we used in [3] and [4].

We analyse and assess the qualified majority (QM) decision rule for the Council of Ministers of the EU, adopted at the Council of the European Union, Brussels, 23 June 2007. This rule is essentially the same as that adopted at the Inter-Governmental Conference, Brussels, 18 June 2004. We compare this rule with the QM rule prescribed in the Treaty of Nice, and the scientifically-based rule known as the ‘Jagelonian Compromise’.

We analyse the projected future evolution of the distribution of voting power and related quantities under the qualified majority decision rule for the Council of Ministers of the EU, prescribed by the forthcoming EU Reform Treaty. Our projections are based on the demographic changes forecast by eurostat [4] for the period stretching from the present to the middle of the 21st Century. We use a method similar to the one we used in [6], [7], [8] and [9].

In a two-tier decision-making system such as the EU Council of Ministers, if the number of constituencies (member-states) is sufficiently large (say, 15 or more), Penrose’s Square-Root rule can be implemented to a high level of approximation by a simple weighted decision rule at the top level (the Council) with any given quota q smaller than the total weight. This leaves one degree of freedom: the value of q as a free parameter, to be determined by some additional condition. I (...) propose to survey and discuss critically the most important considerations for fixing this value – some of which have actually been used by theoreticians or practitioners: efficiency, transparency, sensitivity (total voting power of citizens), mean majority deficit, giving certain ‘privileged’ coalitions blocking status. Some of these considerations are reasonably compatible; others less so. Some kind of compromise is clearly needed. But which? This is essentially a political matter; but a political decision ought to be made in a theoretically enlightened way. (shrink)

I propose a twofold classification of the main considerations underlying the choice of an election procedure: political criteria on the one hand, and social-choice criteria on the other. I formulate political dichotomies, each combination of which narrows down the choice of procedure to a sub-class of the class of all procedures. I discuss what social-choice theory has to offer in each of these.